Truth Tables for 2-Input & 3-Input Circuits

A Truth Table is a comprehensive mathematical table that displays the output state of a digital logic circuit for every possible combination of binary inputs.

1. Determining the Size of a Truth Table

For any digital circuit with $n$ distinct binary inputs, the number of unique input combinations is given by the formula:

• 1 Input ($A$): $2^1 = 2$ rows (0, 1).
• 2 Inputs ($A, B$): $2^2 = 4$ rows (00, 01, 10, 11).
• 3 Inputs ($A, B, C$): $2^3 = 8$ rows (000 through 111).
• 4 Inputs ($A, B, C, D$): $2^4 = 16$ rows.

2. Standard Binary Ordering for 3-Input Tables

In Cambridge examinations, truth table rows must always follow standard ascending binary sequence:

Row 1:  0 0 0  (Binary 0)
Row 2:  0 0 1  (Binary 1)
Row 3:  0 1 0  (Binary 2)
Row 4:  0 1 1  (Binary 3)
Row 5:  1 0 0  (Binary 4)
Row 6:  1 0 1  (Binary 5)
Row 7:  1 1 0  (Binary 6)
Row 8:  1 1 1  (Binary 7)

The Alternation Pattern Trick:

To write an 8-row table quickly without skipping combinations:

• Rightmost input ($C$): alternates every row: 0, 1, 0, 1, 0, 1, 0, 1.
• Middle input ($B$): alternates in pairs: 0, 0, 1, 1, 0, 0, 1, 1.
• Leftmost input ($A$): alternates in blocks of four: 0, 0, 0, 0, 1, 1, 1, 1.

3. Using Intermediate Columns for Complex Circuits

When tracing a multi-gate circuit such as:

Candidates should always construct intermediate columns for each sub-expression before computing the final output $X$:

4. Methodical Evaluation Strategy

1. Calculate column $P$ ($A \text{ AND } B$):

• Only rows where both $A=1$ and $B=1$ produce $1$ (Rows 7 and 8). All other rows are $0$.

1. Calculate column $Q$ ($\text{NOT } C$):

• Flip each value in column $C$ ($0 \rightarrow 1$, $1 \rightarrow 0$).

1. Calculate final output $X$ ($P \text{ OR } Q$):

• Check columns $P$ and $Q$. If either column contains a $1$, write $1$; only write $0$ if both are $0$.